On decay and blow-up of solutions for a nonlinear Petrovsky system with conical degeneration

Abstract

AbstractThis paper deals with a class of Petrovsky system with nonlinear damping $$\begin{aligned} w_{tt}+\Delta _{\mathbb{B}}^{2}w-k_{2} \Delta _{\mathbb{B}}w_{t}+aw_{t} \vert w_{t} \vert ^{m-2}=bw \vert w \vert ^{p-2} \end{aligned}$$ w t t + Δ B 2 w − k 2 Δ B w t + a w t | w t | m − 2 = b w | w | p − 2 on a manifold with conical singularity, where $\Delta _{\mathbb{B}}$ Δ B is a Fuchsian-type Laplace operator with totally characteristic degeneracy on the boundary $x_{1}=0$ x 1 = 0 . We first prove the global existence of solutions under conditions without relation between m and p, and establish an exponential decay rate. Furthermore, we obtain a finite time blow-up result for local solutions with low initial energy $E(0)< d$ E ( 0 ) < d .